From Plurality Rule to Proportional Representation
Selim Jürgen Erguny
July 2008
Abstract
I consider the decision of a parliament that might change the electoral system for the forthcoming
elections from plurality rule to proportional representation. Parties are o¢ ce-motivated. They care
about winning and about the share of seats obtained. I consider two di¤erent scenarios of how parties
in the government share the spoils of o¢ ce: Equally or proportionally to their share of seats. If the
government is formed by a single party and parties expect that each party will obtain the same share
of votes in the next election the electoral rule will never be changed. That is, for a change to occur
the government should be formed by a coalition. I …nd that a change is more likely to occur when the
number of parties is larger and also when the spoils of o¢ ce are shared equally among the members
in the governing coalition. I extend these results to analyze the decision of a change from a less
proportional rule to a more proportional one.
KEYWORDS: Electoral systems, Plurality, Proportional Representation, Coalitions
JEL CLASSIFICATION: D72, H10
I am grateful for the help and valuable comments of my thesis supervisor, Enriqueta Aragones. I would also like to
thank Carmen Bevia, Oriol Carbonell and Jordi Massó for their helpful comments.
y Universidad de Granada, Departamento Teoría e Historia Económica, Facultad de Ciencias Económicas y Empresariales,
Campus de la Cartuja, 18011, Granada (Spain), [email protected]
1
1
Introduction
For political parties whose main objective is to win the elections, that is, to be a part of the government,
there are two main decisions that might be important in order to achieve this goal. The …rst one is their
campaigning activities and the policy promises made to the electorate before the election, because those
might a¤ect the decision of voters at the time of casting a ballot. Secondly, parties might make strategic
constitutional choices in order to increase their future bene…ts. This last question is of great importance
as it is noted by Lijphart (1992) who states that "among the most important -and, arguably, the most
important-of all constitutional choices that have to be made in democracies is the choice of electoral
system, especially majoritarian methods vs proportional representation...". This paper aims to focus on
decisions of political parties of this second type, that is, on their choice of the electoral rule. Naturally, a
choice of this type might have di¤erent e¤ects on the welfare of parties and voters. Therefore, the optimal
choice of an electoral rule might be di¤erent for di¤erent objectives. This paper assumes that parties,
when deciding whether to change the electoral rule, are only considering their own interests. That is, I
assume that parties do not consider voters’evaluation of rules when they take their decision.
I analyze a model of electoral system change where the parliament might decide to change the electoral
system for the forthcoming elections. A simple de…nition of an electoral system would be: "Given a set
of votes, an electoral system determines the composition of the parliament (or assembly, council and
so on)".1 More broadly, as de…ned by Bogdanor (1991), an electoral system can be analyzed in three
dimensions: (1) The method of calculating the votes, or in other words the "electoral formula" (Rae 1971)
where plurality, majority and proportional representation are the main ones; (2) the district magnitude,
that is, the number of representatives or parliamentarians elected in each district, and lastly (3) the
degree of choice a voter can face. This paper focuses mainly on the …rst of these three aspects, the
electoral formula. The structure of the model build up, also refers implicitly to the second aspect.
I consider a situation where the current electoral rule under which the parliament is shaped, is plurality
rule. Under plurality rule, which is sometimes also denoted as First-Past-The-Post (FPTP) system the
winner of the election is the candidate who obtains the highest amount of votes among all candidates.
Plurality rule is generally used in single member districts although there are some exceptions as it is
the case of the election of the Electoral College members in the US (Blais and Massicotte 2002) or in
Mauritius where the legislators are elected from multimember districts (Lijphart 1999). Plurality rule
is being used in countries such as the US, UK, Canada, India and other small ex-British colonies. One
1 Gallagher
and Mitchell (2005) p.3.
2
important characteristic of plurality rule is that it tends to lead to over-representation of larger parties
and under-representation of smaller parties. Since in most of the cases in each district there is a single
winner, a party has to come …rst in a district to win the seat in this district. Clearly, for a party who
has less support it is much less probable that it comes …rst in a certain district compared to a party with
much higher support. Thus, a small party that collects a signi…cant amount of votes spread out among
di¤erent districts, may obtain a very reduced (or zero) amount of seats. This is why plurality rule tends
to under represent smaller parties. As a striking example consider the 1974 British election results2 . The
two larger parties, Labor and Conservatives, obtained 39.3% and 35.8% of the total votes respectively,
whereas the third party, the Liberals, obtained 18.3% of votes. While Labor obtained 50.2% of the seats
and the Conservatives 43.6% the Liberals only obtained 2% of the seats. That is, the largest party was
overrepresented as it obtained more than half of the seats with a vote share of less than 40% whereas the
third party faced a large degree of under-representation as it obtained only 2% of the seats with about
one …fth of the total votes.
Although there exists a huge variety of alternative electoral systems, I assume that the parliament,
given that the proposal for electoral system change reaches the required level of support from its members,
might only decide on the switch from plurality rule to proportional representation rule. Indeed, Colomer
(2005) counts 37 changes during the last century from plurality/majority rules to proportional/mixed
rules among which are the changes occurred in Germany in 1918, Norway in 1919, New Zealand in 1993
and Japan in 1994.
Proportional Representation is used in multimember districts and as described by Taagepera and
Shugart (1989) it "refers to electoral laws that use some mathematical formula for the allocation of seats
to parties in approximation to vote shares". Currently, most of Western European countries use di¤erent
types of Proportional Representation. Proportional Representation rules di¤er in their quotas, that is
the amount of votes obtained by a party which would be worth a seat. One of the main proportional
representation formulas is the Hamilton-Hare exact quota, where a vote share of 1=M , where M is the
district magnitude is worth one seat and the remaining seats are distributed according to the largest
remainders, which is used in Denmark and Costa Rica (Norris 1997). Another formula is the WebsterSainte Lagüe formula where the quota is 1=M + 1 and the remaining seats are given to parties who reach
half of the quota. Another commonly used formula is the Je¤erson-d’Hondt quota of series of divisors,
where the quota for party i is
vi
s+1
where vi is the vote share of party i and s is the amount of seats party
i has been assigned so far. In each round, the party that has the highest quota at the moment is assigned
the corresponding seat. Each of these formulas leads to a di¤erent degree of proportionality. Consider
2 Source:
http://www.election.demon.co.uk/
3
the following example taken from Colomer (2004) where there are four parties (W , X, Y , Z), the district
magnitude is 6 and there are 100 voters. The votes obtained by each party are W = 40, X = 30, Y = 20
and Z = 10 respectively. The seat allocation for each party under these three rules are shown in the
following table where it can be seen that the d’Hondt rule over represents the highest voted party since
party W obtains half of the seats although it only obtains 40% of the votes:
Seat Allocation
Hamilton-Hare
Exact quota: 100/6 =16.6
Jefferson-d’Hondt
Sufficient quota
Webster-Sainte-Laguë
Modified quota: 100/6 +1 =14
By Quota
By Largest Remainders
Total
Total
By Quota
By Half Quota (=7)
Total
W
2
Y
1
2
X
1
1
2
3
2
2
2
1
1
2
2
1
1
Z
0
1
1
0
0
1
1
For the Hamilton-Hare rule the quota is 100=6 = 16:6 as the district size is 6. Therefore, W gets
2 seats and X and Y one seat. The remaining two seats are assigned to the parties with the highest
remainders which are X with a remainder of 13.4 (30-16.6) and Z with a remainder of 10. For the
Webster-Sainte Lagüe formula the quota is 100=6 + 1 = 14. So, W and X get 2 seats and Y one seat.
Since Z’s vote share reaches the half quota (=7) it is given the remaining seat. In the Je¤erson-d’Hondt
quota of series of divisors method the …rst seat is given to the largest party, W . Now the quota of W
becomes 20. So, the second seat is given to the party with largest quota, namely X. Then, the quota of
X becomes 15 and the next two seats are given to W and Y as they have the highest quota, namely 20.
Then, the quota of W becomes 40=(2 + 1) = 13:3 and the quota of Y becomes 10. So, the next seat is
given to X as it has the highest quota and its new quota is 30=(2 + 1) = 10. So, the last seat is given to
W as it has the highest quota.
Another important aspect that contributes to the degree of disproportionality is the district size. As
argued by Lijphart (1999), the degree of proportionality increases with the district size, i.e. the number
of seats available in a district. "For instance a party representing a 10 percent minority is unlikely to
win a seat in a …ve-member district but will be successful in a ten-member district"3 . Empirically the
district size varies from country to country and region to region.
In the following analysis, I assume that under proportional representation each party obtains a share of
seats equal to the share of votes obtained in the elections. This could be considered as "ideal proportional
3 Lijphart
(1999), p.152.
4
representation" which might need very large district magnitudes and can be thought as an approximation
to the case where the whole country constitutes a single electoral district electing a large enough number
of representatives. An example of this case would be Israel and the Netherlands in which the whole
country is a single electoral district with the size of 120 and 150 members respectively. In this model, I
assume that by switching from plurality rule to proportional representation a country would also switch
from single member districts to a single nationwide district.
For plurality rule there does not exist a formula which would give the share of seats of parties taking
as input only the nationwide share of votes each party obtains without taking as input their share of votes
in each district separately. The only possibility is to …nd a theoretical approximation of the seat/vote
ratio of parties in a parliament under plurality. A widely known analytical tool used as an approximation
for plurality rule in single member districts is the so-called "cube law" which was …rst proposed by Parker
Smith, a British mathematician, in 1909. This law states that for the two major parties, the ratio of
their share of seats obtained under plurality rule is approximately the cube of the ratio of the share of
votes obtained. It was found that the "Cube Law" was …tting well to British election results at that
time. This formula was extended by Qualter (1968) to include more than two parties and it was applied
to Canadian election results between 1921 and 1965 where the …t of the "law" was satisfactory. In my
analysis, I assume that the distribution of seats in the parliament will be in line with the prediction of
the "cube law".
That is, for plurality I assume that "cube law" applies whereas for proportional representation I
assume "ideal" proportionality. Later, I relax the assumption on the vote to seat to transformation (cube
law) and I assume that the ratio of the share of seats of any two parties obtained under plurality rule is
the m-th power of the ratio of their share of votes obtained where m is strictly bigger than one.
The aim of this paper is to show under which conditions the parliament of a country would decide to
switch from plurality rule to proportional representation and more generally from an electoral rule that
is not proportional to a more proportional one. A party would be in favor of a change in the electoral
rule only if it would increase its future payo¤s. It can be argued that a party might have two di¤erent
kinds of incentives to favor a change in the electoral rule. First, the new electoral rule might increase
that party’s amount of representation in the parliament, that is, its share of seats. The representation
of a party depends on the amount of votes it and its opponents obtain. Therefore, a party would accept
or reject an electoral system change proposal in accordance with its expected share of votes in the next
elections. Secondly, the new electoral rule might increase a party’s probability of being a member of
the new government that will be formed after the electoral system change. I assume that parties have
5
lexicographic preferences on forming part of a government and on the share of seats they obtain. A
party’s principal aim is to form part of a government with the highest share of seats possible. If it cannot
form part of a government its objective would be simply to obtain the highest share of seats possible. As
stated before parties build their decisions upon their expected share of votes.
Clearly, the probability of forming part of a government does not only depend on the share of seats
obtained but also on the underlying process of coalition formation. Therefore, I consider two di¤erent
speci…cations on how the spoils of o¢ ce will be distributed among coalition members. First, I assume
that coalition members share the spoils of o¢ ce equally and then I assume that they share them in a
manner proportional to the share of seats of each coalition member.
An important aspect of this analysis is the rule used to decide on the change of the electoral system.
I assume that a certain threshold of votes in favor of the change by parliamentarians is needed. I …rst
consider the case where this threshold is the simple majority and then I generalize it to consider any
threshold larger than absolute majority. In the real world di¤erent countries have di¤erent thresholds.
For instance, in France up to 1985 the threshold to change the electoral rule was simple majority and
in 1985 the Socialist government switched from two-round majority to PR as it was in its interest. One
year later the right-wing coalition reestablished the previous rule (Tsebelis 1990). Hungary is an example
of countries that have a threshold larger than the absolute majority where the threshold is a two-thirds
majority (Benoit 2004). I do not consider the possibility that a popular referendum is needed, as it is
the case in Ireland (Benoit 2004).
I …rst consider a political environment with only two parties where parties expect to obtain the same
share of votes in the next election. The main result I obtain is that for any threshold equal or higher
than simple majority, the electoral rule will never be changed. This is so because the larger party is in
o¢ ce and it is overrepresented. Therefore, it is never in the interest of the larger party to change the
rule. Since, the threshold to change the rule is assumed to be at least the simple majority no change is
possible. On the other hand, the opposition party would bene…t from a change in the electoral system as
it would increase its share of seats yet it is never able to get it done. If the assumption on the expectation
of the future share of votes is relaxed, and parties are allowed to have any expectation then if the larger
party’s share of seats is larger then the threshold the electoral rule will be changed if and only if this party
expects to lose the next election. If the share of seats of the larger party is smaller than the threshold
the electoral rule will be changed if and only if both parties expect to lose the next election.
Then, I consider an electoral environment with three parties and assume that parties expect that each
party obtains the same share of votes in the next election. If the government is formed by a single party,
6
then the electoral rule is not changed as the party in power is always against the change. The smallest
party is always in favor of a change, whereas the preferences of the second largest party depend on the
share of votes of his opponents.
In a three party environment when the government is formed by a coalition and the spoils of o¢ ce are
shared equally, the electoral rule change will occur if and only it is in the interest of the second largest
party whose preference depends also on the share of votes of his opponents. This result holds whether the
second largest party forms part of the government or not. As in the case of a single party government, it
is never in the interest of the largest party to change the rule. The smallest party, on the other hand, is
always in favor of a change. If spoils of o¢ ce are shared proportionally, however, a change never occurs
as the two largest parties are always better-o¤ under plurality.
Finally I consider the case of four parties and I assume that parties expect that each party obtains the
same share of votes in the next election. As in the previous cases, when the government is formed by a
single party the rule is not changed. The second largest party still plays a key role in some cases. If spoils
of o¢ ce are shared proportionally, for certain distributions of vote shares a change in the electoral system
that is approved by a absolute majority leads to di¤erent possible coalition government candidates. If
this is the case, the change occurs irrespective of whether the second largest party is underrepresented
or not.
In the literature there exist at least two papers that share similar characteristics with my analysis.
The …rst, set up by Benoit (2004) is a general model of electoral system change. He assumes that the
parties’ objective is to maximize their share of seats, and he gives some real-life examples of electoral
system change and discusses some empirical implications of his model. My analysis could be considered
as an application of this general model to some concrete situations which allows me to solve the model
explicitly and obtain concrete results. Benoit’s theory predicts that the electoral rule will be changed
when a coalition of parties, who have su¢ cient power to change the rule, exists such that each party in
this coalition would gain more seats under the new rule. My model, on the other hand, suggests that,
the situation described by Benoit would not necessarily lead to a change if the probability of being part
of the government for one of these parties is negatively a¤ected. Moreover, the possibility of obtaining a
higher share of seats is not a necessary condition for a party to favor a change.
Boix (1999) analyzes the electoral system change from plurality/majority rule to proportional representation as a result of the entry of new voters (assumed to be leftists) and a new party (socialist) at the
turn of the 20th century in the Western world. He …nds that a change in the electoral rule occurs if and
7
only if it is in the interest of the ruling rightist parties. My model instead is intended to characterize
the cases in which the decision of electoral system change will be taken without assuming any signi…cant
change in the underlying political situation of a country, that is, where there is no threat of new parties
and where parties do not expect huge changes in their share of votes.
The outline of the paper is as follows. In the next section I build up the model. In Section 3,
I describe some general characteristics and results of the model which will be used during the whole
analysis. In Section 4, I analyze the model for two parties. In Section 5, I analyze the case of single
party governments for more than two parties. In Section 6, I analyze the model for three parties and in
Section 7 for four parties under coalition governments. In Section 8, I discuss some implications of the
model and in Section 9, I analyze some extensions of the model. In the last section, I reach some general
conclusions and describe the future research to be done with this model.
2
Model
There exist a number of parties, p. The set of parties is denoted by P = f1; 2; ::; pg. Each party j has
already competed in the past election and obtained a certain amount of votes vj , where vj 2 [0; 1]; 8j 2 P
p
P
and
vj = 1 where vi
vj for i < j. According to the share of votes obtained, the electoral rule
j=1
determines the share of seats of each party in the parliament. The share of seats of party j when rule
p
P
k is applied is denoted by skj 2 [0; 1]; 8j 2 P and
skj = 1 where k represents the electoral rule being
j=1
used, either plurality rule or proportional representation respectively. That is, k 2 fP L; P Rg.
The important aspect here is how votes are transformed into seats. Clearly, this transformation is
a result of the electoral rule used. I assume …rst that, under plurality rule this transformation would
be according to the "cube law". It states that if two parties (a and b) obtain vote shares of va and vb
respectively, then the ratio of parliamentary seats will be
L
sP
a
L
sP
b
= ( vvab )3 . This formula can be applied to
more than two parties in the same manner. That is, for the case of p parties,
Then, I relax this assumption and assume that
si
sj
L
sP
i
L
sP
j
= ( vvji )3 , 8i; j 2 P .
= ( vvji )m , 8i; j 2 P where m > 1. When the electoral
rule is switched form plurality rule to proportional representation, I assume that each party’s share of
R
seats is equal to its share of votes obtained in the election, that is, sP
= vj , 8j 2 P . As stated before,
j
this transformation could be considered as the ideal proportional representation.
Moreover, I assume that parliamentary members are taking their decisions in line with the interests
of their parties rather than their individualist interests of being reelected or of being a member of the
future governments. That is, there exists full party discipline in the decision of electoral system change.
8
Parties care about being in government and the share of votes they obtain. The total amount of
o¢ ce spoils shared among the governing parties is …xed and equal to U . As stated before I consider two
di¤erent speci…cations z (z 2 f1; 2g) on how o¢ ce spoils are shared among parties. I denote the utility
obtained by party j from forming part of a certain government under speci…cation z as Uj;z . Under
the …rst speci…cation (z = 1), I assume that parties forming part of the government share the spoils of
o¢ ce equally, i.e. each party j forming part of the government receives Uj;1 = U = jCj where C is the
governing coalition. Under the second speci…cation (z = 2), I assume that parties forming part of the
government share the spoils of the o¢ ce proportional to their share of votes i.e. a party i forming part
si U
of the government receives Ui;2 = X
where C is the governing coalition.
sj
j2C
These two speci…cations described above might lead to more than one possible winning coalition. If
there exists more than one possible winning coalition, I assume that each of these coalitions has equal
probability of being formed. The utility party j obtains under speci…cation z, Uj;z , takes a di¤erent value
!
p
for every di¤erent winning coalition that contains party j. Thus I de…ne a vector U j;z 2 R2 1 where
2p
1 is the total number of possible coalitions given p parties and leaving out the empty set. Notice
!
that the vector U j;z has a component for each possible coalition, but party j only obtains a positive
pay-o¤ for those winning coalitions of which it is a member. Therefore, only those components of the
vector that correspond to a winning coalition of which party j is a member, will take a positive value,
whereas the remaining components will take the value 0. Similarly, qj denotes a vector of dimension
2P
1 representing the probability of formation of a certain governing coalition that includes party j.
Again, qj has a component for each possible coalition, but only those components of the vector that
correspond to the probability of formation of a winning coalition of which party j is a member will take
a positive value, whereas the remaining components will take the value 0. Then, if we multiply these
!
vectors ( ( U j;z )T qj ) we obtain the expected utility from being in the government for party j.
Parties care not only about being in government but also about the share of votes they obtain. That
is, the utility of party j obtained from its seats is f (sj ) where f (sj ) is assumed to be an increasing
function in sj . I assume that parties have lexicographic preferences. That is the primary objective of
a party is to maximize its expected utility of forming part of the government. Thus, when comparing
two di¤erent electoral rules, a party will …rst compare the expected utility it obtains from being in the
government under each rule, and it will choose the rule from which it derives a higher expected utility.
In case of indi¤erence, the party will choose the rule that would provide it a larger share of seats. In
other words, a party would prefer P R to P L i¤ 1) its expected utility of o¢ ce spoils is higher under P R
or 2) in case it expects the same utility of o¢ ce spoils under both rules then it prefers P R to P L if its
9
seat share is higher under P R.
Moreover, I assume that in order to form a government, a coalition must have more than half of the
seats in the parliament, that is, I rule out the possibility of minority governments. I also assume that the
parliament consists of an odd number of seats in order to avoid having to deal with ties.
Once a government is in power, the government, the opposition, or some members of the government
together with some parties of the opposition might decide to change the electoral rule for the forthcoming
elections. The threshold of share of votes in the parliament needed to change the electoral rule is denoted
1
2.
by T and I assume that T
During the whole analysis, if not otherwise stated, I assume that T is
the simple majority. Moreover, I assume that if a party is indi¤erent between the two electoral rules, it
always opposes the change.
Since parties are o¢ ce-motivated and self-interested, the electoral rule will be changed only if it is
in the interest of at least the absolute majority of the parliament in terms of their expected utilities.
Parties’expected utilities depend on their expected share of votes. Therefore, …rst of all, parties should
have expectations about their own and about their opponents’future share of votes. I denote party i0 s
e
e
e
e
expectations by the vector vie = (vi1
; vi2
; :::vip
) where vij
denotes the share of votes that party i expects
p
P
e
party j would obtain and where
vij
= 1 for all i 2 P . First, I assume that parties expect that
j=1
each party gets the same share of votes in the forthcoming election as they got in the last election, i.e.
vie = (v1; v2 ; :::vp ) for any i 2 P . Later, I relax this assumption and allow for di¤erent expectations over
share of votes.
3
Preliminary Results
This section aims to describe some general characteristics of the model and to obtain some general results
under plurality both under the "cube law" and also for any 1 < m in general which would hold for any
number of parties. The …rst result is as follows:
L
L
Lemma 1: If vi > vj , then sP
> sP
i
j .
Proof: If vi > vj i.e.
vi
vj
sP L
> 1 we have that siP L = ( vvji )m > 1. Therefore, if vi > vj we have that
j
L
L
sP
> sP
i
j . #
That is, the cube law suggests that the share of votes are transformed monotonically into share of
seats. Trivially, this result also holds for PR as we have m = 1. We can also de…ne a party’s share of
seats under the plurality rule in terms of the share of votes of all parties. The result is as follows:
10
vm
L
i
Lemma 2: Under plurality rule sP
=X
p
i
vjm
for any i 2 P .
j=1
L
sP
j
L
sP
P
Proof: Since
v
v2 m
L v1 m
sP
p (( vp ) + ( vp ) + ::: + (
vjm
L
that sP
=X
j
vjm
v
L
L
= ( vPj )m for any j 2 P , that is sP
= ( vPj )m sP
and
j
P
vp 1 m
vp )
L
+ 1) = 1 implying that sP
=
p
p
X
L
sP
= 1, we have that
j
j=1
1
v
v1 m
v2 m
) +( vp
) +:::+( pvp 1 )m +1
( vp
. So we have
for all j 2 P . #
j2P
v3
L
i
So, under the cube law we would have sP
= X
p
i
vj3
for any i 2 P . Notice that for PR we have
j=1
R
that sP
= vi for all i 2 P . We can also …nd some general results on which party(ies) would be
i
underrepresented or overrepresented. The result is as follows:
Lemma 3: i. Under plurality rule party 1 is always overrepresented and party p is always underrepresented.
ii. Under plurality rule, if party i is underrepresented, then any party j with j > i and i; j 2 P is also
underrepresented.
iii. Under plurality rule, if party i is overrepresents, then any party j with j < i and i; j 2 P is also
overrepresented.
Proof: i. We know that
p
X
L
sP
j
j=1
v1
v2
we have
L
sP
2
=
p
X
L
vj = 1. Suppose that sP
1
L
sP
1
L
sP
2
v1 , then since
j=1
< v2 . So, for a similar reasoning
L
sP
3
< v3 and so on... Then,
p
X
L
sP
j
j=1
L
L
Contradiction. Therefore, sP
> v1 . Now, suppose that sP
p
1
L
PL
we have that sP
p 1 > v2 . So, for a similar reasoning sp 2 > vp
vj .
j=1
v
v
j=1
L
Contradiction. Therefore, sP
< vp .
p
<
p
X
= ( pvp 1 )m > pvp 1 ,
p
p
X
X
PL
and so on... Then,
sj >
vj
vp . Since
2
L
sP
p 1
L
sP
p
= ( vv12 )m >
ii. Under plurality rule, for parties i and j with j > i we have that
j=1
L
sP
i
L
sP
j
= ( vvji )m >
vi
vj .
L
So, if sP
< vi
i
L
sP
i
L
sP
j
= ( vvji )m <
vi
vj .
L
So, if sP
> vi
i
L
we must have sP
< vj .
j
iii. Under plurality rule, for parties i and j with j < i we have that
L
we must have sP
> vj . #
j
That is, we have that, under plurality rule the largest party will always be overrepresented and the
smallest party will always be underrepresented. If party i is underrepresented than all parties whose
share of votes are smaller than party i’s are also necessarily underrepresented. We can also describe when
11
a party will be underrepresented under plurality rule in terms of its and its opponents’ share of votes.
The result is as follows:
Lemma 4: Under plurality rule party i is underrepresented if
p
X
vjm > vim
1
.
j=1
L
Proof: If party i is underrepresented, we have that sP
< vi . From, Lemma 2, this implies that
i
p
X
m
vi
< vi which can be rearranged as
vjm > vim 1 . #
p
X
vjm
j=1
j=1
So, under cube law, party i is underrepresented if
p
X
vj3 > vi2 . Notice that, trivially, under PR each
j=1
party is "ideally" represented.
4
Two party model
In this section I analyze the case where there are only two parties and the electoral rule currently used
is plurality. The share of votes obtained of these two parties are v1 and v2 respectively. First, I assume
that vie = vi 8i. That is, each party expects to obtain the same share of votes in the next election. In
this case, would it be in the interest of one or both parties to change the electoral rule and would they
be able to implement it? The …rst result shows that the electoral rule will not change for any threshold
larger than the absolute majority:
Proposition 1: If vie = vi 8i, then for any T
1
2
the electoral rule will not be changed.
L
L
Proof: From Lemma 1 we know that if v1 > v2 we have that sP
> sP
1
2 . From Lemma 3 we know
L
L
that the smallest party is underrepresented under plurality. So, we have sP
> v1 > v2 > sP
and party
1
2
1 has the majority and will form the government. So, party 1 would be against electoral rule change since
R
L
under PR, sP
= v1 < sP
1
1 . Therefore, the electoral rule will never change. #
However, the next proposition shows that it would always be in the interest of the smaller party to
change the electoral system.
Proposition 2: If vie = vi 8i, the smaller party would always be in favor of switching from plurality
rule to PR.
L
R
= v2 > sP
Proof: From Proposition 1 we know that sP
2
2 . Therefore, the smaller party, party 2,
would be in favor of an electoral rule change since it increases its share of votes. #
12
These results imply that, if parties expect to get the same share of votes in the forthcoming election,
then the system would never change from plurality rule to proportional representation.
For all the results above, it was assumed that vie = vi for all i 2 P . How would these results be
a¤ected if we relax this assumption and let each party have any kind of expectation about their share of
votes in the next election? Notice that, since only two parties are competing, the expectations of a party
about its future share of votes necessarily imply its expectation about the share of votes of its opponent.
The following proposition states the results:
Proposition 3: For any T > 21 , the electoral rule will switch from plurality if and only if either
L
(1) sP
> T and v1e <
1
L
(2) sP
< T , v1e <
1
1
2
1
2
for any v2e or
and v2e <
1
2
Proof: From Lemma 1 and Proposition 1, we know that if v1 > v2 , party 1 would have the majority
L
L
in parliament and sP
> v1 > v2 > sP
1
2 . So, the electoral can only be changed if it in the interest of
L
L
party 1. If sP
> T , then party 1 can change the electoral rule by its own. If sP
1
1
T then both parties
should be in favor of an electoral system change. If v1e > 12 , party 1 would expect to obtain the majority
in the next elections which implies that it would expect to win the next elections and obtain more seats
under plurality rule than under PR. So, it would be against a change. Therefore, for a change to occur
party 1’s expectation should be v1e <
rules. So, if
L
sP
1
> T and
party. What if v1e <
1
2
v1e
<
1
2,
1
2,
given that v1e =
1
2,
party 1 would be indi¤erent between both
party 1 would change the rule alone for any expectation of the other
L
but sP
< T ? Then, the electoral rule can be changed i¤ both parties support
1
the change. For the same reasoning as for party 1, party 2 would favor a change if v2e < 12 . #
The proposition states that if the larger party’s share of seats is smaller than the necessary threshold,
the electoral rule will be changed only when both parties expect to lose the forthcoming election. However,
it is very unlikely to occur that both parties expect to lose the election. If the share of seats of the larger
party exceeds the threshold then it is enough that the larger party expects to lose the forthcoming election.
This result implies that under some conditions there might occur changes in the electoral rule for two
party competition which seems contradictory to the reality, as for example in the US, where two parties
compete, the electoral rule has not been changed. However, the results are based on the assumption that
parties can have any expectation about the future which is also not that realistic, as they would certainly
depend also on previous results. Moreover, the model at hand assumes that the change of rule has no
cost for parties in terms of loss of credibility or the e¤ort to pass the change which are certainly factors
that would make a change more di¢ cult. As argued by Benoit (2004), a party that changes the electoral
rule too frequently might be discredited due to the manipulation of the rules for its own interest.
13
In the analysis above I have not made an explicit assumption about the vote to seat transformation
under plurality. As it can be seen the results hold for the "cube law" and any m > 1 in general.
5
Single party government
In this section I consider the case where the government is formed by a single party under plurality rule.
I consider directly the more general case with m > 1 rather than the cube law and do the analysis for
any threshold higher than the absolute majority. From Lemma 1, we know that if vi > vj we have that
L
L
sP
> sP
since the votes are transformed into seats monotonically. Therefore we can obtain directly
i
j
the following result which holds for any number of parties:
Proposition 4: If vie = (v1; v2 ; ::; vp ) 8i and the government is formed by a single party then for any
1
2
T
the electoral rule will not be changed.
L
Proof: If the government is formed by a single party (party 1), we have sP
> 0:5. Since parties
1
expect to get the same share of votes in the forthcoming elections, party 1 will be against any change
L
R
as from Lemma 3, sP
> v1 = sP
1
1 . Therefore, the opposition parties will never be able to change the
system as their share of seats is necessarily smaller than the simple majority.#
That is, when the government is formed by a single party, for any degree of disproportionality and
any threshold higher or equal to the absolute majority the electoral rule will not change. In the next two
sections I will analyze more thoroughly the case of three and four parties. Knowing that, for a change
to occur we need a coalition government it would be wise to describe when this would happen. The
following proposition describes the case of three parties under the cube law:
Proposition 5: If the vote share of the leading party in the elections is smaller than 0:387, the
government will be formed for sure by more than one party. If the vote share of the leading party is
between 0:387 and 0:5 then the government is formed by a single party if and only if v13 > v23 + v33 .
Proof: If the share of votes of the three parties are v1 > v2 > v3 under plurality we must have
sk1
L
> sk2 > sk3 (k 2 fP L; P Rg). From Lemma 2, we know that sP
=
1
government,
L
sP
1
=
v13
v13 +v23 +v33
>
1
2
i.e.
v13
>
v23
+
v33
v13
.
v13 +v23 +v33
To have a one party
should be satis…ed subject to the constraints
v1 > v2 > v3 and v1 + v2 + v3 = 1. So, whenever this inequality holds, the largest party can form
the government alone. It can be shown that the lower bound of v1 satisfying the inequality above can
be obtained when v2 = v3 . In this case, v2 = v3 =
v13
>
(1 v1 )
4
3
1 v1
2 .
which would be satis…ed if v1 > 0:386488. #
14
Therefore, the inequality takes the form of
That is, when the vote share of the largest party is above 0:387, which should be expected to happen
quite often for three parties, it might well be the case that it gets more than half of the seats and forms
the government by its own. Notice that, the higher v1 (given that it is between 0:387 and 0:5) the more
it is probable that the largest party would obtain the majority. For a given value of v1 between 0:387 and
0:5, the closer v2 and v3 are, the higher the probability that the largest party would obtain the majority.
A similar analysis for the case of four parties is shown in the next proposition:
Proposition 6: If the vote share of the leading party in the elections is smaller than 0:3247, the
government will be formed for sure by more than one party. If the vote share of the leading party is
between 0:3247 and 0:5 then the government is formed by a single party if and only if v13 > v23 + v33 + v43 .
Proof: If the share of votes of the four parties are v1 > v2 > v3 > v4 we know from Lemma 1 that we
L
PL
L
L
L
> sP
=
> sP
have sP
> sP
4 . From Lemma 2, we know that s1
1
2
3
L
government, sP
=
1
v13
v13 +v23 +v33 +v43
>
1
2
v13
.
v13 +v23 +v33 +v43
To have a one party
i.e. v13 > v23 + v33 + v43 should be satis…ed subject to the constraints
v1 > v2 > v3 > v4 and v1 + v2 + v3 + v4 = 1. So, whenever this inequality holds, the largest party can
form the government alone. It can be shown that the lower bound of v1 satisfying the inequality above
can be obtained when v2 = v3 = v4 . In this case, v2 = v3 = v4 =
the form of v13 >
(1 v1 )3
9
1 v1
3 .
Therefore, the inequality takes
which would be satis…ed if v1 > 0:3247. #
Notice that, the higher v1 (given that it is between 0:3247 and 0:5) the more it is probable that the
largest party would obtain the majority. For a given value of v1 between 0:3247 and 0:5, the closer v2 ,
v3 and v4 are, the higher the probability that the largest party would obtain the majority.
6
Three party model
In this section, having already discussed before that under a one party government there would be no
change in the electoral rule I consider now a model of electoral system change for three parties where the
government is formed by a coalition. As before, they compete under plurality rule, having the opportunity
to switch to proportional representation. I assume that vie = (v1; v2 ; v3 ) for all i 2 P . I analyze the two
speci…cations for the utility functions of parties separately. First, I consider the case where government
parties share the utility obtained from forming part of the government equally.
6.1
Electoral System Change Under Equally Shared Spoils
If the spoils are shared equally, the government will be formed by the least number of parties possible.
Therefore, C1 (1, 2), C2 (2, 3), C3 (1, 3) would be the three candidates to form the coalition. If a change
15
in the electoral rule occurs, this does not a¤ect the candidates for forming the government. That is,
before and after the change the set of coalitions that might form the government would be the same.
Now we can proceed to analyze when a change would occur. The …rst thing one should notice is that
under equally shared spoils if the government is formed before and after the change by the same parties,
the utility they obtain from being part of the government will not be a¤ected as it does not depend on
the relative size of each party in the coalition. The following proposition shows that for three parties
when the government is necessarily formed by a coalition and the spoils of o¢ ce are shared equally, the
electoral rule will be changed from plurality rule to proportional representation if and only if it is in the
interest of the second largest party. It also shows that it is always in the interest of the smallest party to
change the electoral rule whereas the largest party would always oppose to a change. The results are as
follows:
Proposition 7: If vie = (v1; v2 ; v3 ) 8i, spoils of o¢ ce are shared equally, T =
1
2
and the government
is formed by a coalition, then the electoral rule will be changed if and only if v13 + v23 + v33 > v22 .
Proof: Given that no party can form the government alone (v1 <
1
2
L
and sP
< 21 ), any coalition of
1
two parties would be a candidate to form government. Given that each of these coalitions can be formed
with equal probability and the relative size of each coalition party does not a¤ect the utility obtained
from forming part of the government, a party will be in favor of an electoral rule change if and only if
L
R
R
L
this change would increase its share of seats. From Lemma 3, sP
> v1 = sP
and sP
= v3 > sP
1
1
3
3 .
Therefore, the smallest party would favor a change and the largest party will be against. So, the electoral
R
L
system change will occur if and only if the second largest party is in favor of it, i.e. i¤ sP
= v2 > sP
2
2 .
R
L
So, when does sP
= v2 > sP
hold? From Lemma 4, we need v13 + v33 + v23 > v22 . #
2
2
The proposition states that if o¢ ce spoils are shared equally the electoral rule is changed if and only
if such a change would increase the share of seats of the two smallest parties. From the analysis above
we can deduce that if the share of votes of the two smallest parties are equal then both of them will be
underrepresented. So, the electoral rule will be changed for sure as the change would increase the share
of seats of both parties. Since any coalition is assumed to be formed with equal probability, the change
might be supported by both coalition partners or by one of the coalition partners and the opposition
party.
An important point to notice is that the assumption on the distribution of o¢ ce spoils leads to the
same set of coalition candidates as the minimal winning coalition theory in number of parties suggested
by Leiserson (1966) does. This theory suggests that the government would be formed by a minimal
winning coalition with the minimum of number of parties in it. A winning coalition is a minimal winning
16
coalition (MWC) if the defection of any of his members (in this model parties) turns the coalition into a
loosing one. This theory is based on the bargaining proposition which states that the smaller the number
of parties in a coalition the easier they will …nd it to reach an agreement.
How would the results change if we would consider a threshold higher than the absolute majority and
if we would assume that
si
sj
= ( vvji )m 8i; j 2 P with 3
m > 1? That is, what would happen for a higher
range of thresholds and a generalization of the cube law? The result is as follows:
Proposition 8: If vie = (v1; v2 ; v3 ) 8i, spoils of o¢ ce are shared equally and
rule will be changed i¤ v1m + v2m + v3m > v2m
1
L
PL
and sP
2 + s3
T . If T
2
3
1
2
T < 23 , the electoral
the electoral rule will never
be changed.
Proof: From Proposition 7 we know that the electoral rule will be changed i¤ it is in the interest
of the second largest party, that is when it is underrepresented. From Lemma 4 we know that this
holds i¤ v1m + v2m + v3m > v2m
favor
L
sP
2
L
sP
+
2
L
+ sP
3
L
sP
< 32 .
3
1
. Since, party 1 is always against a change and party 3 is always in
L
T should be satis…ed for a change. However, since sP
>
1
So, if T
2
3
1
3
for sure, we have that
the rule will never be changed as party 2 and party 3 won’t have enough
votes to pass the change. #
The proposition states that if T
2
3,
the electoral rule will never be changed as the sum of share of
seats of the two smallest parties will necessarily be smaller than the threshold. On the other hand, if the
threshold is between
1
2
and
2
3,
then the electoral rule will be changed if and only if the second largest
party is underrepresented and the sum of share of votes of the two smallest parties is larger than the
necessary threshold. So, as before, a change occurs if and only if it is in the interest of the second largest
party.
The minimal winning coalition theory in number of parties is a re…nement of the more general minimal
winning coalition theory (MWC) proposed by von Neumann and Morgenstern (1953). However, for three
parties the predictions of the two theories coincide. Therefore, if we were to do the above analysis under
MWC theory we would reach exactly to the same results.
6.2
Electoral System Change Under Proportionally Shared Spoils
If the spoils of o¢ ce are shared proportionally to the share of seats of the coalition members, the government would be formed by the coalition of the smallest total weight of seats possible. Therefore, we would
have a unique candidate, namely, C1 (2, 3). So, even when a change occurs, the government will still be
formed by those two parties. Under this speci…cation the utility obtained by a party of forming part of
17
the government may not be the same under both rules as the relative size of each party in the coalition
determines the utility obtained by each coalition party and the change in the electoral rule might change
the relative size of coalition partners. So, assuming that the government is formed by a coalition, when
would a change occur? I consider directly the more general case for any 3
1
2.
m > 1 and any T
The
result is as follows:
Proposition 9: If vie = (v1; v2 ; v3 ) 8i, spoils of o¢ ce are shared proportionally and the government
is formed by a coalition, then for any T
1
2
the electoral rule will never change.
Proof: Under this speci…cation the coalition is formed by party 2 and party 3. Party 1 would be
against a change as it is overrepresented. What happens with party 2? Under plurality its utility from
forming part of the government is
Why? If it is true than,
v2m
v2m +v3m
L
sP
2
L
PL U
sP
2 +s3
v2
v2 +v3
=
v2m
v2m +v3m U
and under PR it is
> 0 which is the same as v2m v3 v3m v2 >
v2m
v2
v2 +v3 U . v2m +v3m
0 i.e. v2 v3 (v2m 1
v2
v2 +v3 .
v3m 1 ) >
>
0 which always holds as v2 > v3 . So, party 2 is always against a change whereas party 3 is always in
favor. Therefore a change never occurs. #
That is, under this speci…cation a change in the electoral rule never occurs for any threshold larger
than the absolute majority. The largest and second largest party would be against a change whereas the
smallest party would be in favor. On the other hand if we would consider the situation where the share
of votes of the two smallest parties are equal than a change occurs for sure as long as T
2
3
as both
parties’utility of forming part of the government will be the same under both rules and from Lemma 3,
both will be underrepresented.
The assumption on the proportional distribution of o¢ ce spoils points the same coalition candidate
as the minimum winning coalition theory proposed by Riker (1962) does. This theory suggests that
the government would be formed by a MWC which has the smallest total weight of seats. Using the
assumption that each party expects to receive a larger share of the payo¤ the greater the weight it brings
to a winning coalition, Riker predicted that minimum winning coalitions would form as they maximize
the expectations of each coalition member. Generically, as in the case of three parties, this theory predicts
a unique coalition. So, the results in this section apply in fact for Riker’s theory.
7
Four Party Model
In this section, I consider a model of electoral system change with four parties. I assume that vie =
(v1; v2 ; v3 ; v4 ) for all i. From the previous analysis, we know that if the government is formed by a single
party the electoral rule will not change as all parties expect to obtain the same share of votes in the next
18
elections. Therefore, the electoral rule can be changed only if the government is formed by a coalition.
As in the case of three parties, the two speci…cations on the distribution of o¢ ce spoils will be analyzed
separately. Before doing so, I obtain some results that would hold under both speci…cations:
Lemma 5: i. Under both plurality rule and PR, C1 (3, 4) and C2 (2, 4) can never be winning coalitions.
ii. Under both rules, C1 (1, 4) and C2 (2, 3) cannot be winning coalitions at the same time.
Proof: i. If party 3 and party 4 form a winning coalition, then for k 2 fP L; P Rg, sk3 + sk4 >
implies that
sk1
+
sk2
<
1
2.
Contradiction, as from Lemma 1
sk1
>
sk3
sk2
and
>
sk4 .
1
2
which
A similar reasoning
applies for C2 (2, 4).
ii. Suppose that party 1 and party 4 form a winning coalition. Then sk1 + sk4 >
So,
sk2
+
sk3
<
1
2
1
2
for k 2 fP R; P Lg.
which means that party 2 and party 3 cannot form a winning coalition. #
From parts i. and ii. we can deduce that party 1 and party 2 or party 1 and party 3 can always
form a winning coalition except the case with v1 = v2 = v3 = v4 which is ruled out as the number of
seats in the parliament is assumed to be an odd number. The reasoning in ii. implies that both C1 (1,
4) and C2 (2, 3) will not be a winning coalition i¤ s1 + s4 = s2 + s3 =
1
2
which is also ruled out by the
assumption stated before. Notice that, we know from Lemma 3 that party 2 and party 3 might be under
or overrepresented depending on the share of votes of all parties.
Having found some common properties that would hold under both speci…cations, in the next section
I analyze when an electoral system change occurs with equally shared o¢ ce spoils.
7.1
Electoral System Change Under Equally Shared Spoils
In this section I consider the case where o¢ ce spoils are shared equally and no party obtains the absolute
majority. First we have to …nd which coalitions would be possible under this speci…cation. Disregarding
the case where v1 = v2 = v3 = v4 and given that the parliament consists of an odd number of seats, we
would have the following cases:
Lemma 6: For four parties, when no party obtains the absolute majority alone and o¢ ce spoils are
shared equally the following possible coalitions might be formed (k 2 fP R; P Lg):
Case 1: If sk1 + sk4 <
Case 2: If
sk1
+
sk4
>
1
2
1
2
then C1 (1, 2), C2 (2, 3), C3 (1, 3)
then C1 (1, 2), C2 (1, 3), C3 (1, 4)
Proof: First of all, we can discard coalitions of three parties as the spoils of o¢ ce are shared equally.
So, we have to …nd the possible coalitions of two parties. If sk1 + sk4 <
and party 3 form a winning coalition. Since
sk1
sk2
19
sk3 ,
1
2,
then sk2 + sk3 >
1
2
so party 2
C1 (1, 2) and C3 (1, 3) would also be winning
coalitions. Notice that there exists no other winning coalition of two parties in this case. If sk1 + sk4 > 12 ,
then instead of party 2 and party 3, party 1 and party 4 would form a winning coalition. The other two
coalitions could still be formed for a similar reasoning as before. #
Compared to the case of three parties, di¤erent coalitions might occur for di¤erent distributions of
share of votes. Moreover, the change of the electoral rule might lead to a change in the possible coalitions.
For instance, under plurality rule we might be in Case 1 whereas with the same share of votes under
proportional representation we might reach Case 2. So, to …nd the conditions under which the electoral
rule might be changed, …rst we should check under which of the four cases a change would be feasible
considering all possibilities in terms of a switch from one case to another and the possibility of staying
in the same case. The analysis is as follows where it is assumed that the electoral rule might be changed
from plurality to PR and no party obtains an absolute majority under plurality4 :
1) From Case 1 to Case 2: Party 1 and party 4 would be in favor of a change as their probability of
being part of the government would increase but party 2 and party 3 would be against as their probability
PL
L
< 21 , so no change occurs.
of being part of the government would decrease. In Case 1, sP
1 + s4
2) From Case 2 to Case 1: Party 1 and party 4 would be against a change as their probability of
L
PL
being part of the government would decrease. So, no change occurs as sP
> 12 .
1 + s4
3) From Case 1 to Case 1: The probabilities of being part of a government will not be a¤ected. From
Lemma 3, we know that party 1 would be against and party 4 in favor of a change. So, if both party 2
and party 3 are underrepresented under plurality a change will occur.
4) From Case 2 to Case 2: For the same reasoning as in the previous case i¤ both party 2 and party
3 are underrepresented under plurality a change will occur.
From the analysis above we obtain that the electoral rule will be changed in only two of the four
possible cases. In both cases the change does not a¤ect the composition of the government. Notice that
di¤erent than the case for two or three parties, it might well be the case that the largest party would be
in favor of a change in the electoral rule. Yet we have still to …nd the exact conditions for these two cases
under which the electoral will be changed. The result is as follows:
Proposition 10: If vie = (v1; v2 ; v3 ; v4 ) 8i, T =
1
2
and spoils of o¢ ce are shared equally, then the
electoral rule will be changed if and only if:
4
X
1
L
PL
i. sP
+
s
>
,
and
vj3 > v22 or
2
3
2
j=1
4 This
implies that no party obtains the absolute majority under PR neither. Notice also that it was assumed that each
possible coalition under a speci…c case has the same probability of occuring.
20
L
PL
ii. sP
< 12 , v2 + v3 <
2 + s3
1
2
and
4
X
vj3 > v22
j=1
Proof: We had obtained before that the electoral rule will be changed i¤ this change does not a¤ect
L
L
the future candidates of coalitions. Therefore, for a change sP
+ sP
>
2
3
v2 + v3 <
1
2
1
2,
L
L
or sP
+ sP
<
2
3
1
2
and
is needed. Moreover, both party 2 and party 3 should be in favor of the change, i.e. they
should be underrepresented. From Lemma 3, we know that if party 2 is underrepresented, then also party
4
X
v23
L
3 is underrepresented. So, it is su¢ cient to have sP
=
<
v
i.e.
vj3 > v22 . If both parties
2
2
v 3 +v 3 +v 3 +v 3
1
are underrepresented we have in the …rst case v2 + v3 >
2
L
sP
2
3
+
4
L
sP
3
j=1
> 21 . #
The results suggest that a change in the electoral rule occurs only if this change does not a¤ect the
set of possible coalitions. However, this is not a su¢ cient condition. As in the case of three parties,
the second largest party should be in favor of a change, that is, it should be underrepresented. If this
happens, then all parties except the largest one will be in favor of the change. As it can be seen from
the possible coalitions, it might well be the case that all parties in the government (for example if the
government is formed by party 2 and party 3 in Case 1), or only the smaller coalition partner (for example
if the government is formed by party 1 and party 2 in Case 2) would be in favor of the change. The
following two examples report possible share of votes for which an electoral system change would occur
for both cases.
Example 1: Suppose that v1 = 0:34, v2 = 0:28, v3 = 0:27, and v4 = 0:11. Under plurality rule
L
L
L
L
we would have: sP
= 0:478, sP
= 0:267, sP
= 0:239 and sP
= 0:016. We would be in Case 1, as
1
2
3
4
L
PL
sP
>
2 + s3
1
2
L
and party 2 would be underrepresented as sP
< v2 . If the rule were changed we would
2
still be in Case 1 as v2 + v3 > 12 . So, for the given share of votes, the electoral rule would be changed.
Example 2: Suppose that v1 = 0:32, v2 = 0:25, v3 = 0:24, and v4 = 0:19. Under plurality rule
L
L
L
L
we would have: sP
= 0:475, sP
= 0:226, sP
= 0:2 and sP
= 0:099. We would be in Case 2, as
1
2
3
4
L
PL
sP
<
2 + s3
1
2
L
and party 2 would be underrepresented as sP
< v2 . If the rule were changed we would
2
still be in Case 2 as v2 + v3 < 12 . So, for the given share of votes, the electoral rule would be changed.
In the analysis above it was assumed that T is the absolute majority and that the vote to seat
transformation was according to the cube law. How would the results be a¤ected if we consider higher
thresholds and allow for any 3 > m > 1? First of all, notice that Lemma 6 still holds, as it does not
depend on the vote to seat transformation, that is, the coalition candidates are the same. Moreover, the
analysis for the possible switches form one case to another is not a¤ected neither as it does not depend
on m as long as m > 1. Taking this fact into account, the result is as follows:
21
L
sP
vi m
i
, 8i; j 2 P where 3 > m > 1
L = (v )
sP
j
j
3
< 4 the electoral rule will be changed i¤:
Proposition 11: If vie = (v1; v2 ; v3 ; v4 ) 8i,
1
2
and T
3
4
the
T
electoral rule will never be changed. If
4
X
L
PL
L
PL
PL
i. sP
> 12 ,
vjm > v2m 1 and sP
> T or
2 + s3
2 + s3 + s4
j=1
L
PL
ii. sP
< 12 , v2 + v3 < 12 ,
2 + s3
4
X
vjm > v2m
1
L
PL
PL
and sP
>T
2 + s3 + s4
j=1
Proof: From Proposition 10, we know that the rule will be changed i¤ it is in the interest of party 2.
So, we have only to change the cubes with m. In that case, the change is supported by parties 2, 3 and
L
L
L
4. So, a change occurs i¤ sP
+ sP
+ sP
> T plus the conditions from Proposition 10 changed with
2
3
4
L
L
PL
PL
m’s. sP
> 0:25 for sure, so sP
< 0:75. Therefore, if T
1
2 + s3 + s4
3
4,
a change never occurs. #
It can be found di¤erent distributions of vote shares such that a change occurs for any
However, as T becomes closer to
close to each other. Given that
1
2
3
4
1
2
T < 34 .
a change occurs only if the share of votes of all four parties are very
T < 34 , as before, a change occurs if and only if it is 1) in the interest
of the second largest party and 2) the change does not a¤ect the coalition candidates.
Notice once again that the speci…cation of utilities considered here lead to the same coalition candidates as MWC in number of parties. If we would simply consider MWC, then the only di¤erence we
would have in Lemma 6 would be an additional coalition candidate (C(2; 3; 4)) in Case 2. However, it
can easily be shown that this additional candidate does not change the analysis in this section. That is,
the implications of this model under MWC for four parties would be the same as those for equally shared
o¢ ce spoils.
7.2
Electoral System Change Under Proportionally Shared Spoils
In this section I consider the case where o¢ ce spoils are shared proportionally and no party obtains
the absolute majority. First we have to …nd which coalitions would be possible under this speci…cation.
Notice that as the o¢ ce spoils are shared proportionally only the winning coalition(s) with the lowest
total share will form. Disregarding the case where v1 = v2 = v3 = v4 and given that the parliament
consists of an odd number of seats, we would have the following cases:
Lemma 7: For four parties, when no party obtains the absolute majority alone and spoils of o¢ ce
are shared proportionally the following coalitions might be formed (k 2 fP R; P Lg):
1
2 then C1 (2, 3)
1
> 2 then i) C1 (1, 4)
+ sk3
Case 1: If sk1 + sk4 <
Case 2: If sk1 + sk4
C2 (2, 3, 4) if sk1 = sk2
if sk1 < sk2 + sk3 , ii) C1 (2, 3, 4) if sk1 > sk2 + sk3 and iii) C1 (1, 4),
22
Proof: Case 1: If sk1 + sk4 < 12 , then sk2 + sk3 > 21 . So C1 (2, 3) is a winning coalition. C2 (p1 , p2 ) and
C3 (p1 , p3 ) would also be winning coalitions but with a higher total share. All winning coalitions with
three or four parties will also have a higher total share than C1 (2, 3). So, only C1 (2, 3) might be formed.
Case 2: If sk1 + sk4 > 12 , then C1 (1, 4) is a winning coalition. We can immediately eliminate C(1, 2) and
C(1, 3), the four party coalition and the three party coalitions including parties 1 and 4 as sk2 > sk3 > sk4 .
The only remaining candidate is C2 (2, 3, 4). From these two candidates the smaller in terms of number
of seats is C(1, 4) if sk1 + sk4 < sk2 + sk3 + sk4 i.e. if sk1 < sk2 + sk3 . So, if sk1 > sk2 + sk3 we have as only
candidate C(2, 3, 4) and if sk1 = sk2 + sk3 we have both. #
Notice that, Case 2 has now three subcases depending on the share of seats of the three largest parties.
Now, as in the previous cases we should consider all possible movements from one case to another given
that the rule might change from plurality to PR where it should be taken into account not only the
change of the probability of winning and the degree of representation of a party but also the relative size
of a party in the government which depends on the electoral rule as this a¤ects the utility obtained from
forming part of the government. From an analysis as in the proof of Proposition 9 we can obtain the
following result that will be used throughout the section:
Lemma 8: i. If parties i and j with i < j form the government under both rules, than party i
obtains a higher utility of forming part of the government under plurality than under PR.
ii. If parties i, j and k with i < j < k form the government under both rules, than party i obtains a
higher utility of forming part of the government under plurality than under PR.
Proof: i. Under plurality party i’s utility from forming part of the government is
vi3
U
vi3 +vj3
holds if
vi3
vi3
vi
i
and under PR it is viv+v
U . v3 +v
3 > v +v . Why? If it is true than, v 3 +v 3
j
i
j
i
j
i
j
vi3 vj vj3 vi > 0 i.e. if vi vj (vi2 vj2 ) > 0 which always holds as vi > vj . So,
o¤ under plurality rule. ii.
L
sP
i
L +sP L U
sP
i
j
vi
vi +vj
=
> 0 which
party i is better-
Under plurality party i’s utility from forming part of the government is
L
vi3
vi3
sP
vi
vi
i
L +sP L +sP L U = v 3 +v 3 +v 3 U and under PR it is v +v +v U . v 3 +v 3 +v 3 > v +v +v . Why? If it is true
sP
i
j
i
j
k
k
i
j
j
i
j
j
i
j
k
vi3
vi
3
3
3
3
than, v3 +v3 +v3 vi +vj +vk > 0 which holds if vi vj + vi vk vj vi vk vi > 0 i.e. if vi vj (vi2 vj2 ) + vi vk (vi2
i
j
j
vk2 ) > 0 which always holds as vi > vj and vi > vk . So, party i is better-o¤ under plurality rule. #
Notice that the …rst part of the Lemma implies that the smaller party would be better-o¤ under PR
and the second part implies that for a coalition of three parties the smallest party will be better o¤ under
PR. Under Lemma 7 and 8, the analysis of a possible change is as follows:
1) From Case 1 to Case 1: The probabilities of being part of a government will not be a¤ected. From
Lemma 3, we know that party 1 would be against and party 4 in favor of a change and from Lemma 8
we know that party 2 is also against. So, no change occurs.
23
2) From Case 2 to Case 1: In all three subcases of Case 2, party 4 forms part of the government but
under Case 1 not. So, party 4 would be against a change. Party 1 would also be against a change since
L
PL
its share of seats would decrease under PR. In Case 2, sP
> 12 . So, no change occurs.
1 + s4
3) From Case 1 to Case 2i: Parties 2 and 3 would be against a change as their probability of being
L
PL
part of the government would decrease. So, no change occurs as sP
> 12 .
2 + s3
4) From Case 1 to Case 2ii or 2iii: Is not possible. Why? If under plurality we are in 1, we have
L
sP
1
L
L
< sP
+ sP
which implies that
2
3
v23 +v33
v13 +v23 +v33 +v43
we should reach 2ii or 2iii, we need v1
v13
i.e. v23 + v33
v13 +v23 +v33 +v43
v13 (v2 + v3 )3 . For v2 ; v3
>
v2 + v3 i.e.
> v13 . If after the change
> 0, (v2 + v3 )3 > v23 + v33 .
So, we have a contradiction.
5) From Case 2i to Case 2i: The probabilities of being part of a government will not be a¤ected.
From Lemma 8 we know that party 1 would be against and party 4 would be in favor. Parties 2 and 3
are in favor if they are underrepresented. Therefore, from Lemma 3 for a change to occur party 2 should
be underrepresented.
6) From Case 2i to Case 2ii: Is not possible. Why? If under plurality we are in 2i, we have
L
sP
1
L
L
< sP
+ sP
and to reach 2ii. we need v1 > v2 + v3 i.e. v13 > (v2 + v3 )3 . So, the argument is the
2
3
same as in 4).
7) From Case 2i to Case 2iii: For the same reasoning as in the previous case, it is not possible.
8) From Case 2ii to Case 2i: Party 1 would be in favor of a change as after the change it would form
part of the government and party 2 and party 3 would be against as they would be out of the government.
What happens with party 4? Assuming the change occurs, under plurality its utility from forming part
L
sP
v43
v43
v4
2
4
.
< v1v+v
L
PL
P L U = v 3 +v 3 +v 3 U and under PR it is v +v U .
v23 +v33 +v43
sP
1
4
4
2
3
4
2 +s3 +s4
3
v4
v4
than, v1 +v4
> 0 which is the same as saying v23 v4 + v33 v4 v13 v4 > 0 i.e.
v23 +v33 +v43
> 0. So, we need v23 + v33 v1 v42 > 0. In Case 2i. we should have v1 < v2 + v3 . So,
of the government is
Why? If it is true
v4 (v23 + v33
v23 + v33
v1 v42 )
v1 v42 > v23 + v33
(v2 + v3 )v42 and it can easily be seen that v23 + v33
(v2 + v3 )v42 > 0. So, party
L
PL
4 is in favor of a change and the change occurs as sP
> 12 .
1 + s4
9) From Case 2ii to Case 2iii: Is not possible. Why? We should have v2 + v3 = v1 and sk1 < 0:5
L
k 2 fP R; P Lg. We know that sP
=
1
L
2v1 . So, sP
=
1
v4 = 1
v13
v13 +v23 +v33 +(1
v13
.
v13 +v23 +v33 +v43
2v1 )3
Suppose that v2 + v3 = v1 . So, we have that
. For v2 ; v3 ; v4 > 0, v23 + v33 + (1
2v1 )3 (given that
L
v2 +v3 = v1 ) would be maximum if v3 = 1 2v1 (the smallest value possible). So, sP
would be minimum
1
L
if sP
=
1
v2
v13
v13 +(3v1 1)3 +(1 2v1 )3 +(1 2v1 )3
v3 , we need 3v1
to 0:5 > v1
1
1
2v1 i.e. v1
0:4 we obtain that
L
sP
1
v13
. Moreover, since
1 3v1 3v12 +12v1 3
3
v
L
sP
= 1 3v1 3v12 +12v1 3 with respect
1
1
L
which can be simpli…ed as sP
=
1
0:4. So, if we minimize
< 0:5 can never hold as the minimum it can obtain is 0:5.
10) From Case 2ii to Case 2ii: Is not possible. Given that the change in 9) is not possible, this change
is not possible neither because it requires a even stronger condition, namely, v1 > v2 + v3 .
24
11) From Case 2iii to Case 2i: Party 2 and party 3 would be against as they would be out of the
government after the change. Party 1 would be in favor i¤ his utility of being part of the government
is higher after the change. Assuming the change occurs, under plurality its utility from forming part of
the government is
true then:
v1
v1 +v4
L
sP
1
1
L
PL U
2 sP
1 +s4
3
1 v1
2 v13 +v43 > 0
=
3
1 v1
2 v13 +v43 U
and under PR it is
which would hold if
v14
+
2v43 v1
3
v1
1 v1
v1 +v4 U . 2 v13 +v43
v13 v4
<
v1
v1 +v4 .
Why? If it is
> 0. This inequality always holds
as v1 > v4 . So, party 1 is always in favor. From Lemma 8, party 4 would be in favor5 . So, the change
L
PL
occurs as sP
> 21 .
1 + s4
12) From Case 2iii to Case 2ii: Is not possible. Why? If under plurality we are in 2iii, we have
L
sP
1
v23 +v33
v13
3
3
= v3 +v3 +v
3
3 i.e. v2 + v3
v13 +v23 +v33 +v43
1
2
3 +v4
v2 + v3 i.e. v13 > (v2 + v3 )3 . For v2 ; v3 > 0,
L
L
= sP
+ sP
which implies that
2
3
we should reach 2ii. we need v1 >
= v13 . If after the change
(v2 + v3 )3 > v23 + v33 . So,
we would have v23 + v33 > (v2 + v3 )3 . Contradiction.
13) From Case 2iii to Case 2iii: It is not possible to reach the same situation under both rules.
L
L
L
Why? Suppose we can reach it. Then we would have both sP
+ sP
= sP
(which implies that
2
3
1
v23 +v33
v13 +v23 +v33 +v43
=
v13
v13 +v23 +v33 +v43
i.e. v13 = v23 + v33 ) and v2 + v3 = v1 . So, we should have (v2 + v3 )3 = v23 + v33
which never holds for v2 ; v3 > 0.
From Case 2i to Case 2i, where the change of rule does not a¤ect the possible coalitions, for a change
to occur the same conditions as for equally shared spoils are necessary. However, we would need as an
additional condition that sk1 < sk2 + sk3 should also hold under both rules. Di¤erent than for the case of
equally shared spoils, now we might have cases where a change in the electoral rule alters not only the
degree of representation of each party but also the coalition that forms the government. A change of that
type occurs if we have a switch from Case 2ii or Case 2iii to Case 2i. As in the case of three parties the
change need not necessarily be supported by all parties that form the government. Formally, the result
is as follows:
Proposition 12: If vie = (v1; v2 ; v3 ; v4 ) 8i, T =
1
2
and spoils of o¢ ce are shared proportionally, then
the electoral rule will be changed if and only if:
L
PL
i. sP
< 12 , v2 + v3 < 12 , sk1 < sk2 + sk3 ( k 2 fP R; P Lg) and
2 + s3
4
X
vj3 > v22 or
j=1
L
L
PL
PR
R
R
L
PL
ii. sP
> sP
< sP
+ sP
or
< 21 , v2 + v3 < 12 , sP
2 + s3
1
2 + s3 , and s1
2
3
L
L
PL
PR
R
R
L
PL
= sP
< sP
+ sP
iii. sP
< 12 , v2 + v3 < 12 , sP
2 + s3
1
2 + s3 , and s1
2
3
Proof: i. In this situation we are initially and also after the change in Case 2i. So, we need that
L
PL
sP
<
2 + s3
5 Notice
1
2
and v2 + v3 <
1
2
to be in Case 2. We also need sk1 < sk2 + sk3 ( k 2 fP R; P Lg) to be in Case
that the utility obtained by party 4 in Case 2iii. would be the same no matter which of the two coalitions would
form.
25
2i. before and after the change. The rule will be changed i¤ party 2 is underrepresented i.e.
4
X
vj3 > v22 .
j=1
ii. and iii. describe the conditions such that a change would move the environment from Case 2ii. to
Case 2i. or from Case 2iii. to Case2i. respectively. So, we need the conditions that guarantee that we
L
L
are in Case 2 which are sP
+ sP
<
2
3
1
2
2ii. or Case 2iii respectively, so we need
1
2
L
sP
3
and v2 + v3 <
L
sP
1
>
L
sP
2
+
and before the change we should be in Case
R
R
R
and sP
< sP
+ sP
respectively. The last
1
2
3
conditions in both cases guarantee that we end up in Case 2i. or in Case 2ii. respectively. #
In all the cases where the electoral rule changes, the smallest party forms part of the government
before and after the change. Example 2 would still be valid for Proposition 12ii. The following two
examples report possible share of votes for which an electoral system change would occur for cases as in
Proposition 12i. and 12iii. respectively.
Example 3: Suppose that v1 = 0:3057, v2 = 0:2539, v3 = 0:2302, and v4 = 0:2102. Under plurality
L
L
L
L
rule we would have: sP
= 0:4299, sP
= 0:2466, sP
= 0:1837 and sP
= 0:1398. We would be in Case
1
2
3
4
L
PL
<
2i, as sP
2 + s3
1
2
L
L
PL
PL
and sP
< sP
< v2 . If the rule
1
2 + s3 ; party 2 would be underrepresented as s2
were changed we would still be in case 2i as v2 + v3 <
1
2
and v1 < v2 + v3 . So, for the given share of
votes, the electoral rule would be changed, where the coalition candidate does not change.
Example 4: Suppose that v1 = 0:3480, v2 = 0:3353, v3 = 0:1640, and v4 = 0:1527. Under plurality
L
L
L
L
rule we would have: sP
= 0:4797, sP
= 0:4294, sP
= 0:0503 and sP
= 0:0406. We would be in
1
2
3
4
L
L
Case 2iii, as sP
+ sP
<
2
3
v2 + v3 <
1
2
1
2
L
L
L
and sP
= sP
+ sP
1
2
3 . If the rule were changed we would be in case 2i as
and v1 < v2 + v3 . So, for the given share of votes, the electoral rule would be changed, where
the coalition candidates are changed.
As it was done for the other two theories of coalition formation, I consider now a threshold higher than
the absolute majority. Moreover, once again, we could analyze the possibility of a change by relaxing the
assumption on the cube law in the same manner as it was done before. Lemma 7 would still de…ne the
coalition candidates as they do not depend on m. Moreover, the result in Lemma 8 would also hold6 .
So, we have to check whether there would be some change in the possible movements. (The points from
1 to 13 analyzed above) It can easily be seen that the arguments in points 1,2, 3, and 5 do not depend
on the value of m.7 So they will still be valid. Now reconsider the remaining ones:
4’) From Case 1 to Case 2ii or 2iii: Is not possible. Why? If under plurality we are in Case 1,
L
L
L
we have sP
< sP
+ sP
which implies that
1
2
3
v2m +v3m
v1m +v2m +v3m +v4m
6 These
7 For
>
v1m
v1m +v2m +v3m +v4m
results can be obtained simply by changing 3 with m in the proof of this Lemma.
points 1 and 5 we know that Lemma 8 holds for any m > 1.
26
i.e. v2m + v3m > v1m . If
after the change we should reach 2ii or 2iii, we need v1
v2 + v3 i.e. v1m
(v2 + v3 )m . For v2 ; v3 > 0,
(v2 + v3 )m > v2m + v3m . So, we have a contradiction.
6’) From Case 2i to Case 2ii: Is not possible. Why? If under plurality we are in Case 2i, we have
L
sP
1
L
PL
< sP
and to reach 2ii. we need v1 > v2 + v3 i.e. v1m > (v2 + v3 )m . So, the argument becomes
2 + s3
the same as in 4’).
7’) From Case 2i to Case 2iii: For the same reasoning as in the previous case, it is not possible.
8’) From Case 2ii to Case 2i: Party 1 would be in favor of a change as after the change it would form
part of the government and party 2 and party 3 would be against as they would be out of the government.
What happens with party 4? Assuming the change occurs, under plurality its utility from forming part
L
v4m
sP
4
L
PL
P L U = v m +v m +v m U and under PR it
sP
2
3
4
2 +s3 +s4
v4m
4
Why? If it is true than, v1v+v
>
0
which is the same as
m
m
m
v2 +v3 +v4
4
m 1
m
m
we need v2 + v3
v1 v4
> 0. In Case 2i. we should have v1 < v2
m
1
v2m + v3m (v2 + v3 )v4
and it can easily be seen that v2m + v3m (v2
L
PL
> 12 .
favor of a change and the change occurs as sP
1 + s4
of the government is
v4m
v4
v1 +v4 U . v2m +v3m +v4m
v4 (v2m + v3m v1 v4m 1 )
is
v2m
+ v3 . So,
+
v3 )v4m 1
+
v3m
<
v4
v1 +v4 .
> 0. So,
v1 v4m
1
>
> 0. So, party 4 is in
9’) From Case 2ii to Case 2iii: Might only be possible if m < 1:58496. Why? We should have
L
v2 +v3 = v1 and sk1 < 0:5 k 2 fP R; P Lg. We know that sP
=
1
v1m
v1m +v2m +v3m +v4m .
Suppose that v2 +v3 = v1 .
v1m
So, we have that v4 = 1 2v1 . So,
= vm +vm +vm +(1 2v1 )m . For v2 ; v3 ; v4 > 0, v2m + v3m + (1 2v1 )m
1
2
3
L
(given that v2 + v3 = v1 ) would be maximum (i.e. sP
minimum) if v3 = 1 2v1 (the smallest value
1
m
v
L
L
possible). So, sP
would be minimum if sP
= vm +(3v1 1)m1 +2(1 2v1 )m . Moreover, since v2 v3 , we need
1
1
1
vm
3v1 1
1 2v1 i.e. v1
0:4. For 0:5
v1
0:4, vm +(3v1 1)m1 +2(1 2v1 )m is …rst strictly increasing
1
L
sP
1
and then strictly decreasing for a given m. Moreover, it is strictly increasing in m. For v1 = 0:5 it takes
the value 0:5 for any m. So, if for v1 = 0:4 it takes a value higher than 0:5 then it never can take a
smaller value. It can be shown that it would take a smaller value if and only if m < 1:58496. We need
L
L
L
also sP
> sP
+ sP
which always holds since if v2 + v3 = v1 we have (v2 + v3 )m = v1m > v2m + v3m .
1
2
3
L
L
PL
Therefore, for sure sP
> sP
1
2 + s3 . So, a change might occur only if m < 1:58496 which is supported
by party 1 as it would enter the government after the change and from Lemma 8 also by party 4.
10’) From Case 2ii to Case 2ii: For a similar reasoning as in the previous case a change would be
possible only if m < 1:58496. But then party 1 (as it is overrepresented under plurality) and party 2
(from Lemma 8) would be against a change. So, no change occurs.
11’) From Case 2iii to Case 2i: Party 2 and party 3 would be against as they would be out of the
government after the change. Party 1 would be in favor i¤ his utility of being part of the government is
higher after the change. Assuming the change occurs, under plurality its utility from forming part of the
government is
true then:
1
2
v1
v1 +v4
L
sP
v1m
1
1
L +sP L U = 2 v m +v m U and
sP
1
4
1
4
v1m
1
2
v1m +v4m > 0 which holds
v1
1
v1 +v4 U . 2
v1m+1 + 2v4m v1 v1m v4 >
under PR it is
if
27
v1m
v1m +v4m
<
v1
v1 +v4 .
Why? If it is
0. This inequality always holds
as v1 > v4 . So, party 1 is always in favor. Party 4 is in favor too as the argument in footnote 5 would
L
PL
also hold for any m > 1. So, the change occurs as sP
> 12 .
1 + s4
12’) From Case 2iii to Case 2ii: Is not possible. Why? If under plurality we are in 2iii, we have
L
sP
1
L
L
= sP
+ sP
which implies that v2m + v3m = v1m . If after the change we should reach 2ii. we
2
3
need v1 > v2 + v3 i.e. v1m > (v2 + v3 )m . For v2 ; v3 > 0, (v2 + v3 )m > v2m + v3m . So, we would have
v2m + v3m > (v2 + v3 )m . Contradiction.
13’) From Case 2iii to Case 2iii: It is not possible to reach the same situation under both rules. Why?
L
PL
L
Suppose we can reach it. Then we would have both sP
= sP
(which implies that v1m = v2m + v3m )
2 + s3
1
and v2 + v3 = v1 . So, we should have (v2 + v3 )m = v2m + v3m which never holds for v2 ; v3 > 0.
As we can see from the analysis above, relaxing the assumption of cube law does not change the
results except for point 9’. For this case, there might occur a change if m < 1:58496 which would never
have occurred under the assumption of the cube law. Formally, the result is as follows:
Proposition 13: If vie = (v1; v2 ; v3 ; v4 ) 8i, T >
m
1
2,
spoils of o¢ ce are shared proportionally and
1:58496, then the electoral rule will be changed if and only if:
4
X
L
PL
i. sP
< 12 , v2 + v3 < 12 , sk1 < sk2 + sk3 ( k 2 fP R; P Lg),
vjm > v2m
2 + s3
1
L
PL
PL
, sP
>T
2 + s3 + s4
j=1
3
4
L
sP
2
and T <
ii.
or
L
L
L
PL
PR
R
R
PL
PL
> sP
< sP
+ sP
> T and T <
+ sP
< 21 , v2 + v3 < 12 , sP
3
1
2 + s3 , s1
2
3 , s1 + s4
L
PL
L
L
PL
PR
R
R
PL
PL
= sP
< sP
+ sP
> T and T
iii. sP
< 21 , v2 + v3 < 12 , sP
2 + s3
1
2 + s3 , s1
2
3 , s1 + s4
2
3 or
< 35
If m < 1:58496, then a change occurs i¤ either one of the four conditions above hold or
L
PL
L
L
PL
PR
R
PR
L
PL
PL
iv. sP
< 12 , v2 + v3 < 12 , sP
> sP
= sP
and either sP
>T
1
2 + s3 , s1
2 + s3
1 + s3 + s4
2 + s3
and T <
5
6
L
PL
if party 3 is in favor of a change or sP
> T and T <
1 + s4
2
3
if it is not.
Proof: The conditions except those of the threshold of points i. to iii. is very similar to those of
Proposition 12. Regarding the conditions on the threshold: i. The same argument as in Proposition 11.
ii. In that case an electoral system change is favored by party 1 and party 4. So, besides the conditions
L
PL
L
L
PL
PL
from before we need sP
> T . Since we have a coalition government sP
< 0:5. So, sP
1 +s4
1
2 +s3 +s4
L
L
is at least 0:5. So, the minimum value that can take sP
+ sP
is
2
3
same amount of votes as party 4. Therefore,
L
sP
1
+
L
sP
4
<
2
3.
1
3
as they have to have at least the
So, for any T higher than this value no
change occurs. iii. In that case an electoral system change is favored by party 1 and party 4. So, besides
L
L
L
PL
L
the conditions from before we need sP
+ sP
> T . For a given sP
+ sP
would be minimum if
1
4
1 , s2
3
L
L
PL
PL
L
PL
v2 = v3 = v4 . Given that sP
= sP
= 1. So, sP
can be maximum
1
2 + s3 , we would have 5s4
1 + s4
3
5.
Therefore for T >
3
5
a change never occurs. As was shown above the change in iv. is only possible if
m < 1:58496. The change is favored by parties 1 and 4 for sure but not by party 2. Party 3 might be
28
L
in favor or against depending on m and the vote shares. From ii. sP
is at least
2
1
6
L
PL
and sP
is at
2 + s3
least 13 . Therefore, we need the conditions above. #
Notice once again, that the assumption of proportional distribution of o¢ ce spoils leads to the same
coalition candidates as the minimum winning coalition theory does. Therefore, we can conclude that the
analysis in this section coincides with the result that we would obtain by considering this coalition theory.
8
Implications of the Results
8.1
Size of Government
An interesting aspect that should be analyzed is how the size of the government is a¤ected if a change
in the electoral rule occurs. When parties expect that each party obtains the same share of votes in the
next election, we have seen that no change occurs for the case of two parties. So, the focus will be on the
case of three and four parties. I compare the size of the government before and after the change for each
case in which a change might occur.
For the case of three parties, under equally shared o¢ ce spoils (MWC in number of parties), the set
of possible coalition candidates do not change even if a change occurs. Under the assumption that each
coalition candidate forms the government with equal probability, the expected size of the government
would be:
(sk1 + sk2 ) + (sk1 + sk3 ) + (sk2 + sk3 )
2
=
3
3
k 2 fP L; P Rg
If a change occurs, the decrease in the seat share of party 1 will be equal to the sum of the increase
in the seat share of party 2 and party 3. Therefore, the expected size of the government will remain
2
3
even if the rule changes. When o¢ ce spoils are shared proportionally (MWC in number of seats), we had
previously found that no change occurs.
If there are four parties, then for equally shared o¢ ce spoils, di¤erent than the case of three parties,
as the following analysis shows, if the electoral rule changes the size of the government goes down. Under
this theory, as was shown in Section 7.1, we have two di¤erent cases in which a change occurs:
L
PL
1) If sP
< 21 , then, given that a change occurs, the expected size of the government under plu2 + s3
L
PL
PL
L
PL
PL
PL
PL
PL
2(sP
(sP
1 +s2 )+(s1 +s3 )+(s2 +s3 )
1 +s2 +s3 )
=
3
3
3 )+(v2 +v3 )
size of the government would be (v1 +v2 )+(v1 +v
= 2(1 3 v4 )
3
L
plurality as from Lemma 3, sP
< v4 .
4
rality would be
29
=
L
2(1 sP
4 )
.
3
Under PR, the expected
which is smaller than the size under
L
L
2) If sP
+ sP
>
2
3
plurality would be
1
2,
then, given that a change occurs, the expected size of the government under
L
PL
PL
PL
PL
PL
(sP
1 +s2 )+(s1 +s3 )+(s1 +s4 )
3
expected size of the government would be
under plurality as v1 <
L
PL
PL
PL
L
3sP
1+2sP
1 +s2 +s3 +s4
1
=
. Under PR,
3
3
(v1 +v2 )+(v1 +v3 )+(v1 +v4 )
1
= 1+2v
which is smaller than the
3
3
=
the
size
L
sP
1 .
Di¤erent than above, if o¢ ce spoils are shared proportionally given that a change in the electoral rule
occurs, the size of the government might go up or down. Consider once again Example 2. Under plurality
the government is formed by parties 2, 3 and 4 and its size would be 0:525. Under PR, the government
would be formed by parties 1 and 4 with a size of 0:51. So, after the change the size of the government
decreases. If on the other hand, we suppose that v1 = 0:3202, v2 = 0:2493, v3 = 0:2252, and v4 = 0:2053
L
L
L
the corresponding share of seats under plurality would be sP
= 0:4799, sP
= 0:2264, sP
= 0:1672
1
2
3
L
and sP
= 0:1265. We would be in the same situation as in Example 2, so a change would occur, under
4
plurality the government is formed by parties 2, 3 and 4 with a size of 0:5201. Under PR, the government
would be formed by parties 1 and 4 with a size of 0:5255. So, after the change the size of the government
increases.
8.2
Degree of the Representation of the Government
A widely used argument by advocates of proportional representation is that under this rule the government
would have a larger support in terms of its total vote share compared to governments formed under
plurality/majoritarian rules. Do the above results con…rm this hypothesis?
The approach I take is to compare the total amount of votes obtained by each party of the government
whenever a change occurs. For the case of three parties, when a change occurs, it does not a¤ect the set of
coalitions who can form the government, therefore we can conclude that although the rule might change,
the degree of the representation of the government will not change. For four parties with equally shared
o¢ ce spoils (MWC in number of parties) there will not be a change in the degree of the representation
of the government as a change in the electoral rule does not a¤ect the set of winning coalition candidates
neither.
Under proportionally shared o¢ ce spoils, a change in the electoral rule might a¤ect the composition
of the government. The previous results show that the government under plurality is formed by parties
L
PL
2, 3 and 4 if sP
<
2 + s3
1
2
L
L
PL
and sP
> sP
1
2 + s3 . If this is the case, a change occurs if v1 < v2 + v3 <
1
2
and under PR the government is formed by parties 1 and 4. So, the degree of the representation of the
government under plurality would be v2 + v3 + v4 which is greater than the degree of the representation
30
of the government under PR (v1 + v4 ) as v1 < v2 + v3 . In the other two cases where a change occurs,
the degree of representation of the government does not change8 . So, although it seems counter-intuitive,
a change in the electoral rule might decrease the degree of the representation of the government and
it does never increase it for the case of two, three and four parties under the model at hand. Under
proportionally shared o¢ ce spoils, parties prefer to form government with smaller parties. So, as it was
shown, this preference might lead to a government with a lower degree of representation. The intuition
behind this observation is that if the o¢ ce spoils are shared proportionally the government is formed
by the parties with the lowest total seat share possible. A change in the electoral rule might lead to
formation of an government with a smaller total seat share and degree of representation.
8.3
E¤ective Number of Parties and Electoral System Change
Colomer (2005) by using data of 219 elections in 87 countries runs a regression on data of electoral results
where the dependent variable is the probability of change and the independent variable the e¤ective
number of parties, and he obtains that the probability of a switch from a majoritarian rule to a more
proportional one would increase as the e¤ective number of parties increases. More concretely, he obtains
that only when the number of e¤ective parties increases to 4, the probability of an electoral system change
rises above half (61%). The de…nition of e¤ective number of parties (ENP) used by Colomer is due to
1
Laakso and Taagapera (1979) where EN P = X
p
. What do the results of our model suggest about
vi2
i=1
the relationship between the ENP and a possible change in the electoral rule?
We know that no change will occur for the case of two parties. For the case of three parties, we need
necessarily a coalition government for a change to occur and the o¢ ce spoils should not be distributed
proportionally. So, we need v1 < 0:5. Therefore, for a change to occur ENP should be de…nitely larger
than 2 (it would be closer to 2 as v2 ! 0:5 and v3 ! 0). For three parties EN P can be at maximum 3.
So, we can conclude that for 2 < EN P < 3 a change may occur as the following example shows:
Example 5: Suppose that v1 = 0:37, v2 = 0:33 and v3 = 0:30, where EN P = 2:81 and the spoils
L
L
= 0:3164 and
of o¢ ce are equally shared. Under plurality rule we would have: sP
= 0:4459, sP
1
2
L
sP
= 0:2377. As party 2 is underrepresented, the electoral rule will change.
3
However, one can not say that for a given number of parties, if a change occurs for a certain EN P ,
that a change would de…nitely occur in every other case with a higher ENP. Consider, for instance,
8 It
does not change neither for the additional case that occurs if m < 1:58496.
31
Example 1 where a change occurs. In that example, EN P can be found as 3.58. Now, suppose that
L
v1 = 0:33, v2 = 0:3, v3 = 0:23, and v4 = 0:14 where EN P = 3:68. Therefore, sP
= 0:3468 and
2
L
sP
= 0:1563. We would be in the same case as in Example 1 but the change will not occur as party 2
3
is not underrepresented even though EN P is higher than in Example 1.
9
Extensions
In this section I consider three extensions of the model for the case of two and three parties. First, I
consider the case where "ideal" proportionality is not achieved under PR. Then I consider a situation
where the vote to seat transformation is di¤erent than as it was until now. Lastly, I relax the assumption
that parties expect to obtain the same share of votes under both rules.
9.1
"Non-ideal" Proportionality
Until now I assumed that under PR the seat share of a party was equal to its vote share. In reality,
however, this is, if it is really wanted, a goal hard to achieve. Generally, larger parties are overrepresented
and smaller parties are underrepresented even under PR. To address to this phenomenon, I assume now
that under PR there exists also a certain degree of disproportionality, i.e.
R
sP
i
R
sP
j
= ( vvji )n for all i; j 2 P with
3 > n > 1. For plurality I assume that the cube law applies, that is, plurality is more disproportional. I
consider the case of two and three parties.
As the seat to vote transformation is monotonic, the largest party will be better-o¤ under plurality
and the smallest party under PR. Therefore, for the case of two parties, as before, a change never occurs
as the larger party forming the government alone would be against a change. Similarly, for any number
of parties, if the government is formed by a single party the rule will never be changed.
For the case of three parties, where the government is formed by a coalition, as the following propositions show, the results are almost identical to the ones obtained before. First consider the scenario where
o¢ ce spoils are shared equally:
1
2 and the government
v2n
v23
v1n +v2n +v3n > v13 +v23 +v33 .
Proposition 14: If vie = (v1; v2 ; v3 ) 8i, spoils of o¢ ce are shared equally, T =
is formed by a coalition, then the electoral rule will be changed if and only if
Proof: Given that each of the coalitions with two parties can be formed with equal probability and
the relative size of each coalition party does not a¤ect the utility obtained from forming part of the
government, a party will be in favor of an electoral rule change if and only if this change would increase
32
its share of seats. We know that party 1 will be against a change and party 3 in favor. So, a change
vn
R
L
occurs i¤ party 2 is in favor i.e. if sP
> sP
which from Lemma 3, holds i¤vn +v2n +vn >
2
2
1
2
3
v23
.
v13 +v23 +v33
#
The key party is still the second largest one. The only di¤erence is that now it is in favor of a change
i¤ it is less underrepresented or more overrepresented under PR. As before, if T >
occurs and if
1
2
<T <
2
3
we need
L
sP
2
+
L
sP
3
2
3
a change never
> T . Notice that the above inequality does not necessarily
hold for any n or any distribution of votes. If we assume that o¢ ce spoils are shared proportionally the
result is as follows:
Proposition 15: If vie = (v1; v2 ; v3 ) 8i, spoils of o¢ ce are shared proportionally and the government
is formed by a coalition, then for any T
1
2
the electoral rule will never change.
Proof: Under this speci…cation the coalition is formed by party 2 and party 3. Party 1 would be
against a change for sure. As a rule change a¤ects the relative size of parties 2 and 3, the one whose relative
L
PL
>
size decreases will be against the change. So, no change occurs as sP
1 + s2
1
2
L
PL
and sP
> 12 . #
1 + s3
That is, as before a change is never possible. One important point to mention is that the characteristics
of the results would not change if we take for plurality m with 3 > m > n rather than the cube law.
9.2
A More General Vote to Seat Transformation
Now I assume that the vote to seat transformation for three parties under plurality is as follows:
( vv21 )m
and
L
sP
2
L
sP
3
=
v2 n
( v3
)
L
sP
1
L
sP
2
=
where m > 1, n > 1 and m 6= n. Notice that this is the most general case possible.
For PR, I assume that ideal proportionality applies. So, from an analysis similar to Lemma 2 we obtain
L
that sP
=
1
v1m v2n m
,
v1m v2n m +v2n +v3n
L
sP
=
2
v1m v2n
v2n
m
+v2n +v3n
L
and sP
=
3
v1m v2n
v3n
m
+v2n +v3n
. From an analysis very
similar to Lemma 3i. we can obtain that party 1 is overrepresented and party 3 is underrepresented.
Therefore, as before, under a single party government the rule will never change.
If we consider equally shared o¢ ce spoils and assume that the government is formed by a coalition,
party 1 will be against and party 3 in favor of a change. Therefore, the key party is as before party 2. A
change will occur i¤ party 2 is underrepresented, i.e. i¤ v2 >
v1m v2n
v2n
m
+v2n +v3n
.
If we consider proportionally shared o¢ ce spoils, the coalition is formed by parties 2 and 3. Party
1 would be against a change as it is overrepresented under plurality. From the same analysis as in
Proposition 9, we obtain that party 2 would also be against a change. Therefore, a change never occurs.
As it can be seen, the results have the same characteristics as before.
33
9.3
Expectations with Changing Vote Shares
Until now I assumed for the case of three and four parties that parties expect to obtain the same share
of votes in the next election whether the electoral rule changes or not. However, it has been argued that
voters who would prefer to vote for one of the smaller parties, end up voting for the "less evil" of the
two leading parties in order not to waste their vote when the electoral rule is plurality. That is, smaller
party would increase their vote share under PR. In order to address to this behavior I assume now that
parties’ expectations of their future vote share would be di¤erent for the two rules and I consider the
case of three parties. If the electoral rule does not change, parties still expect to obtain the same share of
votes as before, i.e. vie = (v1; v2 ; v3 ) for any i 2 P . If the electoral rule switches to PR however, I assume
that parties expect that the share of votes of parties 1 and 2 will decrease whereas the share of votes
of party 3 will increase, and that all parties have the same expectations, which is not too unrealistic as
in the real world there exist quite accurate poll results. That is, vie = (v10 ; v20 ; v30 ) for any i 2 P where
v1 > v10 , v2 > v20 and v30 > v3 . Moreover, I assume that the order of parties in terms of their share of
votes does not change, that is, v10 > v20 > v30 . Notice that, if the government is formed by a single party
the rule will not change.
Consider the case where o¢ ce spoils are shared equally and the government is formed by a coalition.
Party 1 who is overrepresented under plurality will be against a change. Party 3, as before, will be in
favor of a change. Therefore, as before, the key party is the second largest one. If party 2 is overrepresented under plurality it will be against a change. So, for a change to occur we need party 2 to be
underrepresented under plurality and that it expects to obtain more seats under PR, i.e. a change occurs
i¤ v20 >
v23
9
.
v13 +v23 +v33
Notice that, the result has the same characteristics as before but it is possible for a
smaller range of vote shares as v2 > v20 .
Now suppose that spoils of o¢ ce are shared proportionally among coalition members where the government is formed by parties 2 and 3. Party 1 would be against a change as its share of seats would
decrease. Now suppose that one of the remaining two parties is in favor of a change. This means that
this party’s relative size in the government increases with the rule change. However, then the other party
should be against a change as its relative size in the government would decrease. Therefore, as before, a
change never occurs.
9 This
result would hold for any m > 1 by changing the cubes with m.
34
10
Discussion and Concluding Remarks
The analysis shows that for three or more parties the electoral rule cannot change if the government is
formed by a single party and parties expect that each party obtains the same share of votes in the next
election. This …nding might help to understand why the electoral rule has never been changed in Britain
where the current rule is plurality. Over the last decades, many times the debate of changing the electoral
rule in Britain has been brought up, yet with no success. If the governments formed in Britain from 1945
up to today10 are examined, we can see that it has never been formed by a coalition of parties.
The results obtained also indicate that the fact that the government is formed by a coalition does
not necessarily imply that the electoral rule will be switched to proportional representation. For the
case of three parties the key party is the second largest one independently of whether it forms part of
the government or not. Considering di¤erent theories of coalition formation, I found that the strategic
choices of parties with respect to a change in the electoral rule are not the same under all these theories.
On the other hand, with a change in the electoral rule, the possible governing coalition candidates are
still the same ones.
For the case of four parties if it is assumed that coalitions are formed according to MWC or MWC
in number of parties (equally shared o¢ ce spoils), the set of governing coalitions in not a¤ected neither.
If it is assumed that the underlying theory is MWC in number of seats (proportionally shared o¢ ce
spoils), then in the cases where the electoral rule might be changed without a¤ecting the candidate(s)
of governing coalitions, the key party is still the second largest party. However, there might well occur
a change for some distributions of vote shares, which alter the candidate(s) for governing coalitions. In
those cases, the second largest party does not play a key role. One interesting point to mention is that
while in the case of three parties the largest party is always against a change and the smallest party
always in favor, in the four party case this is not always the case. That is, the largest party although it
is overrepresented under plurality might be in favor of a change as it might enter the government after
the change while it would out of the government under plurality. Similarly, the smallest party, always
underrepresented under plurality, might be against a change when it forms part of the government as the
change might lead to a coalition of which it does not form part. The analysis shows that for four parties
a change is possible for a higher range of thresholds compared to the case of three parties.
Clearly the results are obtained by maintaining some assumptions. One of these assumptions was
about the assignment of seats according to the share of votes under plurality. First, I assumed that the
1 0 Source:
Web page of Richard Kimber, http://www.psr.keele.ac.uk/
35
share of seats are obtained according to the "cube law" where the share of seats ratio of two parties is the
cube of the share of votes ratio. Laakso (1979) argues that it would be more appropriate to take a ratio
of 2.5 rather than 3 as it …ts better the British election results. On the other hand, Maloney et al. (2001)
examining election results of six countries where non-proportional rules are used (Australia, Canada,
France, New Zealand, UK and US) argue that the ratio would be between 2 and 3 for the two major
parties. When this assumption was relaxed, I obtained that for the case of two and three parties (for
all three theories of coalition formation) the general characteristics of the results do not change because
they do not depend on the magnitude of the ratio. In the case of four parties, the results do not change
neither for the theories of MWC and minimum winning coalitions in number of parties. In the case of
minimum winning coalitions in number of seats, however, if the degree of disproportionality is su¢ ciently
low, a change occurs under a wider range of conditions.
Another assumption made in the analysis is that under proportional representation each party’s share
of seats equals its share of votes. Obviously, none of the PR rules used today in elections can reach
total proportionality. Yet, as the analysis above shows that relaxing this assumption for the case of three
parties does not change the results. However, in the case of four parties, the degree of proportionality of
the alternative rule might play an important role. This aspect needs a further analysis.
One further step that should be taken in the analysis is relaxing the assumption that parties expect
that each party obtains the same share of votes in the future elections for four parties as it was done for
two and three parties. It can easily be argued that a party’s expectation would depend on his past share
of votes and on the electoral rule at hand. It is a well-known argument that plurality rule leads to higher
degree of strategic voting compared to proportional representation. Therefore it would be wise to take
the expectations of parties in a more sophisticated manner as a function of their past share of votes and
the electoral rule. I believe that this analysis would lead to interesting implications.
36
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